Exegesis:

kai exeplēssonto epi tē didachē autou ‘and they were astonished at his teaching.’ For ekplēssomai cf. on 2.48.

didachē ‘teaching’ may refer to the act or to the content. Here it probably includes both, but with an emphasis on content, rather than on form.

hoti en exousia ēn ho logos autou ‘for his word was with authority.’ logos is best understood as “what he said” (Translator’s New Testament, New English Bible), because exousia would never apply to the act of speaking apart from what is said and apart from the person who speaks. en exousia therefore refers implicitly to Jesus himself as well.

exousia ‘authority,’ cf. on v. 6. Here it has not the concrete sense but refers to the authority which Jesus’ words command. German translations render Vollmacht, which is more than “authority” and less than “absolute power” but refers to an authority derived from and bestowed by a higher power, here by God.

Translation:

They were astonished, see on “wondered” in 1.21; 2.10. The term occurs also in 4.6 (which see), 36; 5.24; 7.8; 9.1; 10.19; 12.11 (plural, referring to persons); 19.17; 20.2, 8, 20; 22.25.

Quoted with permission from Reiling, J. and Swellengrebel, J.L. A Handbook on the Gospel of Luke. (UBS Handbook Series). New York: UBS, 1971. For this and other handbooks for translators see here . Make sure to also consult the Handbook on the Gospel of Mark for parallel or similar verses.